A variant of d'Alembert functional equation on monoids

In this paper, we determine the complex-valued solutions of the functional equation $$f(x\sigma (y))+f(\tau (y)x)=2f(x)f(y)$$ for all $x,y\in M$, where $M$ is a monoid, $\sigma$: $M⟶M$ is an involutive automorphism and $\tau$: $M⟶M$ is an involutive anti-automorphism. The solutions are expressed in terms of multiplicative functions, and characters of $2$-dimensional irreducible representations of $M$.